Randomized approaches to accelerate MCMC algorithms for Bayesian inverse problems

Markov chain Monte Carlo (MCMC) approaches are traditionally used for uncertainty quantification in inverse problems where the physics of the underlying sensor modality is described by a partial differential equation (PDE). However, the use of MCMC algorithms is prohibitively expensive in applications where each log-likelihood evaluation may require hundreds to thousands of PDE solves corresponding to multiple sensors; i.e., spatially distributed sources and receivers perhaps operating at different frequencies or wavelengths depending on the precise application. We show how to mitigate the computational cost of each log-likelihood evaluation by using several randomized techniques and embed these randomized approximations within MCMC algorithms. The resulting MCMC algorithms are computationally efficient methods for quantifying the uncertainty associated with the reconstructed parameters. We demonstrate the accuracy and computational benefits of our proposed algorithms on a model application from diffuse optical tomography where we invert for the spatial distribution of optical absorption.